Unlike last week, today I came well prepared for these questions, thanks to Ted Alper of the Stanford Math Circle. I had reached out to the 1001 Circles Facebook group for help with this problem, and Ted came to my rescue with another example that better illustrates this characteristic of the algorithm.* The students did the new example, saw what was going on, and then we moved into another discussion of the practical applications of this theoretical model. Last week talked about matching doctors to hospitals. This week we discussed an article (by the Royal Swedish Academy of Sciences) that further explained why this algorithm was Nobel worthy and what others did with the algorithm to extend it.

One thing that we talked about doing last week was continuing the proofs behind this algorithm. We ended up running out of time. I suggested that the students watch Dr. Rhiel's second Numberphile video ("the math bit") to see the proofs, just in case we run out of time in the course. I hope to return to these next week, but I also hope an interested student or two might want to watch the videos and lead a discussion of the proofs with the group.

In our final 15 minutes, we revisited our project of creating individual college-admissions algorithms. I told the students that my goal is to put their algorithms into an excel spreadsheet so that we can run various hypothetical students through it. “Cool!” was the biggest reply.

Finally, I invited students to let me know after class if they wanted to present any of their own algorithms, or those they’re interested in, or proofs of algorithms (see above, hint, hint!) in our final session in a few weeks. I hope some do!

Rodi

*Ted Alper and Benjamin Leis both responded to my post in 1001 Circles and gave me help. I just love that the math circle community is so supportive. Ted recommends that interested teens read the original article by Gale and Shapley, “College Admissions and the Stability of Marriage.” I've read it and would encourage this too. During each class session, I talk with the students about what I've posted in these online reports. Often students want to read more about the things we talk about, so please forward them these reports so they have the links. Thanks, parents!

]]>“The arc of the moral universe is long, but it bends toward justice.”

-- Martin Luther King, Jr.

This week we dug into Chapter 2 of the text titled "Drawing the Color Line". In this chapter, Zinn lays the groundwork for a discussion about racism in the colonies and frames the content around the question of if racism is a natural human tendency or a manufactured institution.

We reviewed the chapter, with a particular focus on the concept of "colonialism" and the histories of Jamestown, John Punch, and Bacon's Rebellion. Though chapter 3 will continue to discuss these in more detail (specifically Bacon's Rebellion), at this point Zinn attempts to drive home the idea that the colonies could not survive without free labor, and attempts to do so were desperate and full of horror and misery. Colonists came to learn that that forcing the people living here already to work for them was too difficult, as the indigenous people were too familiar with the land and were too able to quickly organize resistance. The colonists also came to realize that the indentured servitude of Europeans was effective but not sustainable, as the Irish and German and other people brought here to work (often against their will) were too familiar with the culture and language of the colonists and could organize effective resistance. The introduction of enslaved people from Africa provided the manageable labor force the colonies required, as the displacement from their homeland and introduction into a foreign culture, alongside a system of brutal indoctrination and a policy of dissolving and scattering tribal and family units, made resistance significantly more difficult. However, it quickly became apparent that the biggest threat to the growing prosperity of the Colonies was the risk of indentured servants, first peoples, and enslaved people joining forces.

We conducted an exercise involving the discussion of 5 incidents where the actions of the disenfranchised people in the colonies threatened the establishment, and I asked the participants to predict what kind of laws the governments would create to counter these incidents. For each incident, we compared the predictions of the participants with the actual laws that were established.

The laws created in response to the incidents were designed to specifically to divide and target people of color. As the population of enslaved people from Africa was growing at incredible rates, laws were purposefully crafted that punished people of color more harshly, that punished white colonists for aiding enslaved African people, that limited and controlled the movement and freedom of people of color (even if they were "free") significantly more than white colonists, and that punished white colonists for engaging in personal relationships with people of color.

As we worked through the incidents, the predictions from the participants shifted from hypothetical laws that would protect the wealth of the elites to hypothetical laws that would alienate and oppress people of color. This was the history that Zinn wanted to focus on in this chapter; that the systematic approach to "drawing a color line" around the people living here would establish an institution of racism that would echo throughout the history of the United States.

A lot of the content in this chapter was disturbing -- graphic descriptions of the life at Jamestown prior to using forced labor, graphic descriptions of the treatment of enslaved people being transported from Africa to the colonies, graphic descriptions of the brutal treatment of indentured servants and enslaved people. I started the program with a discussion of the MLK quote at the top of this page, a quote he created from the words of Theodore Parker, as a means of reminding the participants that studying history can be uncomfortable and cause negative feelings, but it is important to consider that perhaps despite the horrible things people do, we continue to move, albeit slowly and maybe too slowly at times, toward a more moral and just place.

I pointed out that in 2012, Ancestry.com published a paper suggesting that John Punch, the “first official slave in the English colonies", was a twelfth-generation grandfather of the first African American to become President of the United States (Barack Obama) and is also believed to be one of the paternal ancestors of the 20th-century American diplomat Ralph Bunche, the first African American to win the Nobel Peace Prize.

**Assigned Reading:** Chapter 3, “Persons of Mean and Vile Condition” addresses the serious class divisions in the English colonies. Zinn argues that the elites used a variety of means to stay in power, including fabricating division between the middle and lower classes and enlisting the support of the middle classes by encouraging a fear of the lower classes. I also asked the particpants to consider this questions and be prepared to discuss their thoughts: Should employment status relate to a person’s rights? Are those with tax-paying jobs deserving of privileged treatment by the government?

**-- Adrian**

I was right; the class was captivated by the problem. But it turned out that the problem was much deeper than I was prepared for. (I shouldn’t have been surprised by this, since I knew that this algorithm eventually led to a Nobel prize for one of its creators.)

Does the algorithm give the same result if you start with the men? S said yes, M said no. Several students tried to work it out in their heads with our current example and quickly agreed with S (that the answer is yes). M wasn’t so sure. She was adamant that we actually do it out. So we did. And it turned out that with our specific example, we got the same result. It looked like S was right. Until I told them that the algorithm can favor the person who proposes. “Why did our example work out the same?” The prevailing student conjecture was that it was something specific about the order of our lists. They were all too much alike. Another conjecture was that our sample group was too small. I admitted to the students that I don’t know the math behind this problem well enough to answer (I only learned it a week ago), but told them I’d try to find out. My plan is not to just give them the answer to this question next time, but to give them more scenarios so they can discover it for themselves.

We got into the math (theorems and proofs) behind the algorithm. I got confused in one of the proofs (a proof by contradiction); I wasn’t sure whether we were doing it right. It is a short but confusing proof. Some students were following, some had no interest. Sometimes the younger students just check out mentally when the mathematics gets too abstract. The older students wanted to figure it out for themselves but the younger ones had gotten off the bus so to speak. I was glad to notice that we were out of time. We had been working on this problem intensely for 75 minutes without even a quick bathroom break. Had we more time, I would have given a break right then. But it was time to leave. So I promised everyone that we’d start with it first thing next time. This would give me time to learn the proof better.

On the one hand, it might seem that this was not a successful math circle considering that I couldn’t answer all of the questions and got confused in a proof. But I posit that seeing me struggle with math was very good for the students. Students can expect their leaders to be walking Googles, and that creates a distance. It can make it hard for students to see themselves as mathematicians or even problem solvers. So when they get to see how everyone struggles with some things in math, it can give them hope. No students were annoyed with me for not knowing the things I didn’t know. I think it motivated some of them to try to figure it out for themselves.

The students did enjoy using the puppets. This is my first time using them with teens and I was happy that they enabled the students to go deeper into the mathematics than they may have otherwise. There were a few times when the mathematical struggle was a little intense, and then someone broke the tension by making their puppet say something funny.

By the way, I didn’t write a full recap of session 3, so here’s an overview. We finished work on the Google Page Rank algorithm. Students were exposed to some new mathematics – probability distribution – and had a lot of questions and a lot of fun. Then students then developed their own college admissions algorithms for hypothetical colleges of their own design:

- Kale University (“for really smart people”)
- Collegiate University of Redundancy
- Universal University
- Monster University
- Redundant School of Redundancy
- TSU/Top Secret University (“This is not its name; no one knows its name.”)
- University of the Underworld (“only for people who are dead”)
- Prison College (“an online school for people serving life sentences”)

Just from the list of schools here, students could already see the dangers of using a single algorithm on a large scale.

To be continued in another session.

Oh, one more thing I wanted to tell you: I’ve been showing students pictures of the mathematicians behind the problems we’ve been working on in this course. It is exciting for them to see so many diverse faces (age, race, gender, and even formality – Emily Riehl’s faculty page shows her playing the guitar in a T-shirt!).

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FIBONACCI

Dr. Lawrence used conventional algebraic notation including variables with subscripts and matrices. I wanted to know how comfortable our students would be with this while also keeping their minds wrapped around the definition of an algorithm, so I put a series of numbers on the board: 0,1,1,2,3,5,8,13,…. (This is not something she did.)

Several students quickly identified this list as the Fibonacci series. I asked

- Is this an algorithm?
- Could the number 4 be on the list?
- What is the rule?
- How can you express the rule symbolically?

We discussed, and ended up with the conventional notation for this on the board. We have such a wide range (6 years*) of age and experience in the group that it didn’t surprise me that this algebraic notion was old news to some students and totally unfamiliar to others. We did a quick calculation or two then I reminded everyone of the big picture in this course: algorithms, their applications, and their misapplications.

WHERE DOES THAT PAGE ORDER IN SEARCHES COME FROM?

The class brainstormed what they knew about how Google comes up with the list order. They didn’t know much, but the rest of the class raised their eyebrows at how much one student knew about browsing in incognito mode. Then I gave Dr. Lawrence’s example and we worked through it. We ended up with a graph theory graph on the board. This is not a traditional coordinate-plane, xy-axis type of graph. This is a graph of a network with edges and vertices.

OUR DIVERGENCE FROM THE PRESENTATION

Dr. Lawrence’s presentation used both graph theory and the notation of a system of linear equations with variables and subscripts. Our discussion was juggling the math in the exact same way as Dr. Lawrence. The graph notation was easy for everyone to follow. The equations, though, were not. Once they were on the board, the most experienced students were smiling and nodding but some of the least experienced wore deer-in-the-headlights expressions. Hmmmmm…. what to do?

Since it seemed that everyone understood what was going conceptually, the only issue was the notation. We had to (A) tell the same story without variables, (B) work through some simpler variable scenarios to aid in comprehension for some students, or (C) keep going with this notation with only some people understanding. I thought quickly about the emotional state of the students. Option (A) would work for everyone if only I knew of a way to do it. Option (B) would leave some students bored, and maybe even resentful of being in a class with people who hadn’t seen this notation before. Option (C) would dig a deeper hole of confusion and maybe even anxiety for some other students.

Fortunately, I got very lucky. First of all, M said, “I don’t understand!” relieving some of the tension in the room. Second, I somehow saw a way to do option (A). Phew! I realized that we could use numerical calculations without variables and mark those results directly on the arrows on the graph.

My 20/20 hindsight tells me I should have anticipated this problem before class and have an alternative approach to the problem in my metaphorical back pocket. But I didn’t. I’m feeling grateful that something occurred to me on the spot. I also wished I had talked to the students about the importance of acknowledging their own feelings/reactions in math. We also could have talked about the different ways people react emotionally to math problems. (One student told me later that working with symbols makes her feel good, that they make her feel smart.)

I am happy that the students got to enjoy the delight of an unexpected mathematical result (ask your children, or watch the video!). If you do watch the video, know that we didn’t get through all of it, and will pick up next time at the part where we come up with the probability distribution and test it.

Looking forward to continuing with this problem next week! I do plan to continue to present the material with the algebraic notation, since familiarity will increase comfort and usability for the younger students and will be respectful to the older students. I expect to face the above pedagogical dilemma again and again. This will be fun! (Those of you who know me will know that I am being serious, not ironic.)

Rodi

*Why, you may ask, do we have such a wide range of ages in one group? The answer is that this wide range insures us enough enrollment to have a big enough group for meaningful and energetic mathematical conversation and collaboration. We have 9 students, which allows for many perspectives and insights.

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You are charged with the mistreatment and murder of thousands, perhaps millions, of Taíno Indians.

This week we conducted a role-playing exercise created by Bill Bigelow, the Rethinking Schools curriculum editor and Zinn Education Project co-director. The exercise is called "The People vs. Columbus, et al.". You can view and download this exercise here.

*“This role play begins with the premise that a monstrous crime was committed in the years after 1492, when perhaps as many as three million or more Taínos on the island of Hispaniola lost their lives. (Most scholars estimate the number of people on Hispaniola in 1492 at between one and three million, some estimates are lower and some much higher. By 1550, very few Taínos remained alive.) Who -- and/or what -- was responsible for this slaughter? This is the question students confront here.” -- Bill Bigelow*

Following the script in the exercises, we divide the participants into 5 groups; each group represented either Columbus, Columbus' Men, King Ferdinand & Queen Isabella, Taínos, or The System of Empire.

Each group had 3 participants. Indictments, which detailed the charges against them, were given to each group to review. They had approximately 40 minutes to prepare their defense.

When we reconvened, each group designated one member of their defense team to serve on the Jury, one member to be the primary spokesperson, and one person to be the secondary spokesperson. This left us with a Jury of 5, and each group represented by two defendants.

As the facilitator, I played the role of prosecutor.

We kept the format of the trial in line with the exercise script: the prosecutor read the indictment out loud to the courtroom, the defendant came forward and read their defense and called any witnesses they wanted from the other groups, then the Jury could ask whatever follow-up questions they wished.

Each of the defendant groups did an amazing job defending themselves, pulling all of the obvious rationalizations you'd expect, but also surprising me with some very creative defenses. For example, when attempting to defend The System of Empire, the defendant stated that "while my system, unfortunately, allows for abuse and atrocities, it does not require them; you still have to choose, on your own, to commit them." I thought that was surprisingly astute.

By working hard to defend each of these groups, the hope was that each group would be examined for its complicity in this crime, and I feel this was most definitely accomplished.

After we completed this process for each of the five groups, the Jury went off to deliberate; their task was to determine guilt, if any, and to what degree (expressed by percentages).

In the end, the Jury found all parties, except the Taínos, guilty as charged; they broked down the guilt by percentage, like this:

- Columbus was found guilty and 60% responsible.
- Columbus' Men were found guilty and 20% responsible.
- The Royals were found guilty and 10% responsible.
- The Taínos were found not guilty and the court apologized for them even being there.
- The System of Empire was found not guilty and 10%responsible.

While perhaps there is no right or wrong answer to what percentage of guilt should be assigned to which group, I suggested that by assigning the bulk of the guilt to Columbus, the Jury was making a statement about if humans have free will, or if we are controlled by our culture and environment, with a nod to the former. Admittedly, this is more of a philosophical discussion, and not specifically part of this program; however the defense of “But I was just following orders!” will come up often as we move through the material, and I want the group to be prepared to consider its validity.

We did not have enough time to explore their responses to last week's take-home question (Should we celebrate Columbus?), and I want to allow that this exercise may have given some of them cause to change their answers, so we'll dive into that first thing next week.

**Assigned Reading:** Chapter 2, "Drawing the Color Line" addresses the African slave trade and servitude of poor British people in the Thirteen Colonies. Zinn writes of the methods by which he says racism was artificially created in order to enforce the economic system. He argues that racism is not natural because there are recorded instances of camaraderie and cooperation between black slaves and white servants in escaping from and in opposing their subjugation.

**-- Adrian**

September 21, 2017 (Week One)

“An interesting mix, capable of beautiful dreams and terrible nightmares.”

-- Carl Sagan

We opened up the program this week by reviewing responses to our first take-home question: What Is America? Most of the participants prepared two to three words, and we listed them on a whiteboard. Here are some of the responses:

- Beautiful
- Multicultural
- Racist
- A Melting Pot
- Divided
- Egotistical
- Hopeful
- Powerful
- Scared
- Brave

We discussed how the United States of America can be all of those things -- and more -- at the same time. We discussed the power of Culture and how it affects how we view events. We discussed how so much of history is a mesh of good and bad things happening simultaneously. We used the examples of how the USA could put a person on the moon without the use of what we would consider "computers" while at the same time committing atrocities in Viet Nam, along with a few other examples.

We then handed out the books; every participant received a new, hardback copy of Howard Zinn's *A People's History of the United States*, and previewed the first chapter. We discussed how Columbus was a "round-earther" at a time when the flat-earthers were dominant and explored concepts like colonialism and economic colonialism, the cultural need for holidays, what nations generally do with an infusion of wealth, and why the "new world" became an obsession of Europe.

**Assigned Reading & Question:** I asked the group to read “Chapter 1: Columbus, The Indians, and Human Progress”, for next week, and to come prepared to discuss their thoughts on the following question: Should we celebrate Christopher Columbus?

**-- Adrian**

This fall, our Tuesday Making and Exploring program is digging into the book *My Side of the Mountain* by Jean Craighead George. *My Side of the Mountain* is an exciting novel about a fifteen year old boy who leaves New York City to live off the land in the wild. In this twelve week class, participants will study the book and participate in wilderness survival and nature studies activities, such as: Foraging, Study of Falcons, Pouch Making, Plant Identification, Cooking with Wild Ingredients, Mapping, and more.

The novel will be divided into sections, one for each week, that participants read before the next class session. To accommodate all reading levels, any method of consuming literature is encouraged. The participants can read it to themselves, it can be read aloud to them, or they may listen to an audiobook. At the beginning of each session we will review what happened in that part of the book. The rest of each class will be an experiential activity based on the reading.

Ages 8-10

Tuesdays 1 PM - 3 PM

12 weeks from SEP 19 - DEC 5

Garden Classroom

Tuition $180

To register please email angie@talkingsticklearningcenter.org

More about the book:

“My Side of the Mountain, written by Jean Craighead George in 1959, is a survivalist story about a boy who runs away from home to live in the Catskill Mountains, and he not only survives but thrives in the wilderness. The story begins with Sam Gribley already in the mountains preparing his humble tree abode for the first snowstorm. He discusses in detail some of the challenges he's faced so far and his fear of the storm and not knowing what will happen after. Then gradually, he talks about his life in New York, his family, and how he came to the Catskills." from a goodreads.com review.

“Sam heads up to his great-grandfather’s property, and there begins his life in the wild. He burns out a cave in a massive Hemlock tree in which to live, and begins to gather and store food, from plants and wild vegetables to apples and small game. Sam also manages to take a baby peregrine hawk from her nest, and raises her as a pet, naming her Frightful. Despite the mountain’s isolation, many tourists and local residents end up passing by. Sam avoids most of them, speaking to only a few, such as Bando, the English professor, who ends up at Sam’s place by accident, having lost his way. Sam spends the autumn continuing to collect food and supplies for the winter. The first snow storm passes, and at Christmas, Bando returns. Sam’s dad also appears for a visit and is incredibly proud of his son.” from bookrags.com

]]>>>>*What is an algorithm?* The students defined it as an input and output with a certain way of solving something. They brought up and asked some questions about the Pandora algorithm (specifically it’s “randomness”), which we will return to during another session.

>>>*What are some common algorithms and how do they work?* We worked through conversion from Fahrenheit to Celsius and also prime factorization. Then I showed the students an article in today’s Philadelphia Inquirer about new Youth Poet Laureate Husnaa Hassim.

>>>*If you wanted to create a website or app that provides its users with a poem every day, how would you be sure to provide poems that your users would like?* The students said they’d ask users to click if they like the poem, watch for trends (feedback!), and survey users ahead of time for 6 things: what’s going on in your life, how was your day, genres/poets they already like/dislike, your religion, your core values, and your style of humor. A discussion emerged about what the right number of questions would be. “What does this have to do with algorithms,” asked the class. You will see, you will see!

>>>*What would you do if you were in charge of a large, urban, struggling school district and you needed to raise student performance?* The students brainstormed and debated. I asked “who is easy to blame?” for student performance. I asked what variables they would consider if they had to rank teachers to weed out ineffective ones. The students said they’d ask kids, sit in and observe, evaluate teaching performance, and perhaps give teachers a sample lesson to teach. They grudgingly said that if money were an issue, they may have to use student testing data, but that they wouldn’t want to. I then shared the story of teacher Sarah Wysocki’s unfortunate experience with the teacher performance algorithm “Impact” described in detail in Cathy O’Neill’s Weapons of Math Destruction (WMD).

>>>*Do you care how the temperature algorithm works?* Only one student raised her hand.

>>>*Do you care how the prime factorization algorithm works?* That same student raised her hand, and a few put their hands up a few inches, but most kept their hands down very low.

>>>*Do you think people care how the Impact algorithm works?* “They should!” declared D, with many nods of agreement. (I’m hoping that by the end of this course, people will not automatically accept mathematical algorithms – whether pure math or applied – on faith.)

>>>*What is your algorithm for packing your lunch?** The students brainstormed a list (which would, of course, vary by student) of the variables they would consider: food availability, hunger, what they’re doing that day, nutrition, taste, portability, quantity, whether sharing, dietary restrictions, and variety. (Just an aside here: I found it interesting that no one mentioned cost. I wonder how this list would differ had parents written it.)

>>>*Could you write down your algorithm so that someone else could pack your lunch for you?* Students weren’t sure. J began to write her algorithm. Others began to discuss potential difficulties: things that can vary day by day, the skill of the person preparing it, mood, birthdays, availability, seasonality, tiredness. We then discussed the difference between a formal and informal algorithm and also what it means for an algorithm to be trustworthy. “So you’re saying that the algorithm has to be dynamic; it has to react to changing conditions?” Most said yes. J, however, posited that her algorithm didn’t have to be dynamic, that it would work every time. M seemed to agree, and began proposing his own algorithm. J read her algorithm to the group. M and J posited that it might be possible to forsee every variable. People started poking holes in their algorithms. “You’re behaving exactly like mathematicians,” I remarked. “One person posits a proof, and other mathematicians try to find its flaws.” The discussion continued. The debate focused on whether an algorithm must be dynamic in order to be trustworthy. This was a debate I definitely didn’t anticipate when I was planning this session, so it was thrilling to watch.

We were then out of time, to be continued next week.

Rodi

*The lunch packing algorithm example is based upon O’Neill’s dinner prep example in WMD.

]]>It's no secret that many kids say they hate math. Perhaps what they are saying is that they hate the way math is taught in the classroom.

For many years now, Talking Stick Learning Center has offered a Math Circle to the local community as one of our supplementary programs.

Our Math Circle has been recognized nationally and continues to be led by Rodi Steinig, a Mt. Airy math educator.

This fall and spring, we will be offering 4 brand new Math Circles. They are exciting, new programs custom designed by Rodi for specific age groups in our unique community.

Registration for all 4 Math circles is now open.

If you'd like your child to participate in this wonderful program, please register as soon as you can, as space does fill up quickly.

**Our Algorithmic Culture**

Designed for ages 13 and up, this math circle will focus on what algorithms are and how do they drive our culture. Thursdays, 3:30-4:45pm, Cope House. 9/7/17 to 10/26/17.

Register

**Embodied Mathematics**

Designed for ages 5-7, this math circle will open students’ minds to the reality that math is about more than numbers. Thursdays, 3:30-4:30pm, Garden Classroom. 11/2/17 to 12/7/17.

Register

**Invariants**

Designed for ages 8-10, this math circle will engage with a variety of tools to understand invariants. Thursdays, 3:30-4:30pm, Garden Classroom. 1/25/18-3/1/18.

Register

**The Platonic Solids**

Designed for ages 10-14, this math circle will explore foundations of geometry with hands-on activities. Thursdays, 3:30-4:30pm, Cope House. 3/22/18 to 5/17/18.

Register

__Our Algorithmic Culture__

Dates: Thursdays, 3:30-4:45pm, 9/7-10/26 (8 weeks, 75-minute sessions, 10 hours total)

Suggested Ages: 13+

What are algorithms and how do they drive our culture? We’ll examine the Google page-rank algorithm, Cathy O’Neill’s National-Book-Award-nominated *Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy*, whether random number generators are really random, the mathematics behind “fake news,” the Euclidian algorithm, and much more. These topics will provide context for a study of the algebra behind algorithms: sequences of instructions that usually involve variables and (algebraic) expressions. These expressions can be organized logically into matrices, programs, flowcharts, etc., to produce solutions to well-defined problems. Then, of course, we will debate the appropriateness of labelling problems “well-defined.” While we’re at it, we’ll delve into some statistics and number theory and then compose some algorithms of our own.

Students should be familiar and comfortable with variables, although the ability to manipulate them is not a prerequisite. While this is a students-only course, interested parents and guardians are invited to participate during the final session.

__Embodied Mathematics__

Dates: Thursdays, 3:30-4:30pm, 11/2-12/7 (5 weeks, no class on Thanksgiving)

Suggested Ages: 5-7

Neuroscience has provided empirical evidence of what we intuitively knew all along: that counting on your fingers enhances learning. The discipline of embodied mathematics employs gesturing and physical interactions with the environment to develop conceptual understanding and to facilitate articulation about mathematical concepts. Year after year, young students come into Math Circle with the idea that mathematics is all about quick computation and nothing else. This course will open students’ minds to the reality that math is about more than numbers and can be explored with more than a computational approach. We’ll use our bodies and surroundings to examine symmetry, 2D and solid geometry, equivalence, measurement, spatial reasoning, and arithmetic computation.

__Invariants__

Dates: Thursdays, 3:30-4:30pm, 1/25-3/1 (6 weeks)

Suggested Ages: 8-10

An invariant is a quantity whose value never changes no matter what you do to the operation under consideration. For example, when you shuffle a deck of cards, the number of cards in the deck remains unchanged. Mathematicians consider invariance one of the most important concepts children need to know as they go through their math educations. In this course, we’ll engage in problem solving, games, flowcharts, storytelling, and a hands-on exploration of the Euler Characteristic to search for and understand invariants.

__The Platonic Solids__

Dates: Thursdays, 3:30-4:30pm, 3/29-5/17 (8 weeks)

Suggested Ages: 10-14

Students will engage in hands-on activities to discover some fundamental principles of geometry. We’ll create and classify the platonic solids as we build with Polydrons. We’ll explore fractals as we attempt to build a 3D Sierpinski Triangle from business cards. We’ll make discoveries about area as we fold paper into squares and triangles. We’ll explore and expand upon Euclidian geometry as we fold more paper. We’ll see math in nature through a look at how ladybugs fold their wings. And in a study of empirical versus logical proofs, we’ll use toilet paper to explore what variables come into play – and how they interact – as we try to figure out the maximum number of times you can fold a piece of paper.

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